Convexity of the reachable set of nonlinear systems under \(L_2\) bounded controls. (English) Zbl 1050.93007

Nonlinear, finite-dimensional, time-dependent control systems described by ordinary differential equations are considered. The main purpose is to formulate sufficient conditions for the convexity of the reachable set for the system with \(L_2\)-bounded control. In the proof of the main results, convexity conditions for the nonlinear image of a small ball in a Hilbert space are extensively used. Moreover, for the special case of linear control systems, convexity of the reachable sets is also discussed. The connections between controllability of the linearized control system and the convexity of the reachable set for nonlinear control system are also considered. Some optimal control problem is analyzed and an illustrative numerical example is presented. Many remarks and comments on convexity of the reachable sets are also given. Similar results for linear systems can be found in the paper [R. E. Kalman, Y. C. Ho and K. S. Narendra, Contrib. Differ. Equ. 1, 189–213 (1963; Zbl 0151.13303)].


93B03 Attainable sets, reachability
93B05 Controllability
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
49K15 Optimality conditions for problems involving ordinary differential equations


Zbl 0151.13303